The most simple and therefore the most inclusive description of the method that I have discovered to achieve both phase and amplitude reconstruction along with super resolution is based primarily on three mainstays: the Rayleigh-Sommerfeld scalar wave diffraction equation, the mathematical law of Parseval and the principle of reconstruction through “iterative error reduction” (Gerchberg, R. W. and Saxton, W. O., 1972, Optik, 35, 237).
Physically, the problem involves a three dimensional instrument having transverse Cartesian coordinates X & Y and an orthogonal Z coordinate along the direction of propagation of the wave front of the scattered radiation. I define two conjugate transverse planes at Xo, Yo and Xd, Yd. The subscripted o (output) plane is upstream of the subscripted d (diffraction) plane with the scattered wave front propagating from output to diffraction plane along the Z axis. To meet the strict requirements of the scalar diffraction equation, the wave front is considered to be null or zero at the output plane except within a moderately small hole say at the origin (Xo=0, Yo=0). Still more physically, the output plane is a total occluding surface not allowing the scattered wave to exist in that plane at any point that is not within the area of the small hole. The diffraction plane is downstream from the output plane by a distance Zd sufficiently large so that the two dimensional wave front there can be expressed as the two dimensional Fourier transform of that same wave front at the output plane upstream. Each point within the volume of space between the two conjugate planes can be located by its unique X, Y and Z coordinates. The wave function at each point has a complex value consisting of two rather than one number. The two numbers are called the amplitude and the phase. In sum then, one has a two dimensional amplitude distribution and a two dimensional phase distribution describing the scalar output wave everywhere in the space from Zo to Zd, and in particular in the output plane. This wave propagates to the diffraction plane at Zd. The distribution in the diffraction plane of the amplitude and phase and therefore of the scattered wave is very different from what it was in the output plane. Mathematically the scattered wave in the output plane has been morphed into a new scattered wave in the diffraction plane by having traveled the distance Zd between the two planes. The output wave has been Fourier transformed to yield the new diffracted wave. Now the essence of the first problem, the Fourier inverse problem becomes clear.
Ordinarily, the Fourier Transform would take the complete (two dimensional complex) wave form in the output plane and transform it to the complete wave form in the diffraction plane. But current technology can only provide a way to measure the amplitude distribution (without the phase distribution) in the diffraction plane easily and with considerably more difficulty, the amplitude distribution (without the phase distribution) in the output plane. The phase information is not measurable and hence the scattered wave has lost a large percentage of the information it carries. Some estimate as much as 80% of the information in the scattered wave is lost here. However, assuming that we know the amplitude distribution alone in both planes, it is possible via the Gerchberg-Saxton (Gerchberg, R. W., 1974, Optica Acta, v. 21, n. 9, 709) algorithm to infer the complete wave function in most cases. Now, assuming that we have the more likely situation, in which only the amplitude distribution is in hand for the diffraction plane, the phase distribution is again able to be calculated by knowing certain other constraints which exist naturally or which have been artificially placed on the wave function in the conjugate output plane. One of these useful constraints might be the boundary of the hole in the occluding screen through which the wave propagates. Mathematically, this boundary defines a so called area of support of the wave function at the output plane. Another constraint might place a known range of values, within which the amplitude and/or the wave phase must be. In fact, the investigator may make his own set of constraints by placing various phase plates (or partially occluding plates) whose specifications are known, within the area of support, and measuring the different diffracted wave amplitudes (without the phase distribution) corresponding to each perturbing filter. An arrangement such as this was studied by me. The computational algorithm (a modified form of the Gerchberg-Saxton algorithm) is given in “A new approach to phase retrieval of a wave front” by R. W. Gerchberg. (Gerchberg, R. W., 2002, J. of Modern Optics, v. 49, n 7, 1185).
The R. W. Gerchberg article shows that the method is always successful in finding the correct wave function with its phase function in both the output plane and the diffraction plane.
The main thrust of this project was to develop a process including data-gathering and processing techniques that would consistently and without fail discover the phase distribution of a wave front. In the course of the work, new thoughts on data gathering evolved to the point where a novel type of intensity picture bearing little or no resemblance to the wave front or its phase distribution was required for mathematical processing to achieve the phase distribution. For convenience these information images have been dubbed phasorgrams. Phasorgrams are recorded in the diffractive plane or in the image plane of an application device. Additional data, while helpful, are not required, thus eliminating the need for data from two Fourier conjugate planes as in the well-known method of Gerchberg and Saxton. The mathematical processing of these data is carried out by an iterative algorithm which also is new. This algorithm, given good data, has not failed to achieve the phase distribution. Contemporary devices have not been designed to measure the new kinds of data required for the process. Therefore, several novel devices are suggested. The new process could have major implications for the phase problem of X-ray crystallography.
A scalar wave front may be represented as a two-dimensional complex function. At present, the amplitude of this complex function is routinely measured but the associated phase or angular distribution is not directly retrievable. The problem that this paper addresses is a new method of inferring the phase distribution from the measured amplitude distribution of the wave front. Many computer simulations have been run using the new method. All of these have been easily successful in discovering the phase distribution
The diffraction theory of Huygens, Young, Fresnel, Kirchhoff, Rayleigh and Sommerfeld referred to in Chapter 3 of Goodman's [1] book provides the basis of the method. Using the paraxial approximation and the restrictions of Fresnel diffraction, Goodman shows that the initial wave front at z=zi which is propagated along the z axis (normal to the wave front), is related to the observed wave front downstream of at zi, at z=zo, by the equation
                              U          ⁡                      (                                          x                0                            ,                              y                0                                      )                          =                              1                          j              ⁢                                                          ⁢              z              ⁢                                                          ⁢              λ                                ⁢                      exp            ⁡                          (                              j                ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                z                            )                                ⁢                      exp            ⁡                          [                                                                    (                                          j                      ⁢                                                                                          ⁢                      k                                        )                                    ⁢                                      (                                                                  x                        o                        2                                            +                                              y                        o                        2                                                              )                                                                    2                  ⁢                  z                                            ]                                ⁢                      ∫                          ∫                                                U                  ⁡                                      (                                                                  x                        i                                            ,                                              y                        i                                                              )                                                  ⁢                                  exp                  ⁡                                      [                                                                                            (                                                      j                            ⁢                                                                                                                  ⁢                            k                                                    )                                                ⁢                                                  (                                                                                    x                              i                              2                                                        +                                                          y                              i                              2                                                                                )                                                                                            2                        ⁢                        z                                                              ]                                                  ×                                  exp                  ⁡                                      [                                                                                            (                                                                                    -                              j                                                        ⁢                                                                                                                  ⁢                            2                            ⁢                            π                                                    )                                                ⁢                                                  (                                                                                                                    x                                0                                                            ⁢                                                              x                                i                                                                                      +                                                                                          y                                0                                                            ⁢                                                              y                                i                                                                                                              )                                                                                            z                        ⁢                                                                                                  ⁢                        λ                                                              ]                                                  ⁢                                  ⅆ                                      x                    i                                                  ⁢                                  ⅆ                                      y                    i                                                                                                          (        1        )            where U(x, y) is the total complex wavefunction in the (x, y) plane normal to the z axis, z is the drift distance between the initial wave front and the observed wave front (i.e. z=zo−zi), λ is the wavelength, the subscript i indicates quantities in the initial wave plane, the subscript o indicates quantities in the observed wave plane and k is the free-space wavenumber in radians per wavelength.
An eminently readable derivation of this diffraction equation (1), and the paraxial and Fresnel diffraction constraints is given in Chapters III and IV of Goodmans book. Equation (1) is the keystone of this entire project. It validates the notion that, assuming full knowledge of a scalar wave front (a two-dimensional complex function of Cartesian coordinates x and y in a plane normal to the z direction of propagation), it is possible to calculate the two-dimensional complex wavefunction at any drift distance downstream. Thus it is possible to pose an inverse question: given only the measured intensity of a coherent monochromatic wave front in an observation plane at coordinate zo along the direction of wave propagation, is it possible to determine the wave front completely (both the amplitude and the phase) at the initial coordinate zi upstream from zo?
Equation (1) shows that the wave front at zo is related to the wave front at zi in a fairly simple way. First, there is a constant factor exp (jkz)/jλz, which is determined by the wavelength and the drift distance between the initial and observation planes. Then there is a quadratic phase factor exp [(jk)(x02+yo2)/2z], which is a function of the wavelength and the lateral position in the observation plane. We note that the phase functions in both of these factors are not visible to current measuring equipment so that the effect of both factors is simply a magnification constant of 1/λz. What is measurable is the amplitude of the Fourier transform of a complex function in the initial plane which may be referred back to the actual complex wave front in the initial plane in a simple manner. The original question then simplifies to the following: given the amplitude distribution of the complex wave front in the observation plane, can one generate its phase function and thereby provide the information to reconstruct the actual wavefunction in the initial plane uniquely?
The answer of course is no. There is just not enough information to solve this inverse problem uniquely. More information is needed.
Many real situations provide the possibility of acquiring additional independent information. This is true because both the initial and the observation planes are available for experimentation. One may measure wave intensity in either or both of the Fourier conjugate planes and one may alter the wave in either of these planes by using well-defined devices or perturbers in them such as lenses, stops or kinoform lenses. By way of example, in an ordinary light camera with a single convex lens, the relationship between the wave front in the back focal plane (the initial plane) and the image plane (the observation plane) is that of Equation (1) with coherent monochromatic light. Again equation (1) would seem to hold for the wave leaving the face of a crystal and its diffraction pattern. These are only two of the many situations in which equation (1) applies and in which both the initial and the observation planes are available for gathering data and for experimentation. Gerchberg and Saxton (GS) [2] took advantage of this possibility by measuring not only the intensity of the wave in the image plane (observation plane) but also the intensity of the wave in the Fourier plane (initial plane) in the electron microscope. The result provided an answer for discovering the complete wave front, both the amplitude and the phase, in a few but not all cases. Kim [3] very recently published a paper in which the initial plane has been partially obscured in order to generate a new synthetic diffraction plane image. These data are used in a GS-type algorithm to solve partially the phase retrieval problem, and then the result is used as a first estimate to improve further the answer using a second diffraction pattern resulting from an unobscured initial plane. In Kim's simulation, he knows that the initial wave is real as opposed to being complex and that the initial wave is always positive. It is implied that this information is used to help to solve this phase retrieval problem.
Our study also creates synthetic images with which to solve the phase retrieval problem. The initial wave will, however, not be limited to a real function and the perturbation in the initial plane will not be limited to a large region which has been set essentially to zero while retaining the rest of the wave front unperturbed.
This project has, for example, employed kinoform binary phase lenses of very different binary phase distribution patterns which were inserted sequentially into the initial plane to change the phase distribution there, with the result that vastly different observation plane waves, each corresponding to a different initial plane lens, were synthesized. Each lens created a new picture or phasorgram. That particular demonstration used six different kinoform lenses to create the six different phasorgrams which are shown in FIG. 1. The pictures shown there do not resemble the diffraction pattern generated by the unperturbed initial plane wave. That pattern would have been constant across the observation plane. There was no observable detail there. The six different phasorgrams in FIG. 1 plus knowledge of the six binary phase kinoform lenses phase distributions were all that was used to solve that phase retrieval problem.
The new algorithmic process for solving the phase problem mathematically is cyclic. The complex function representing each phasorgram in the image plane is estimated and then these estimates are used to generate a single new estimate of the complex wavefunction in the initial plane. This new initial plane complex function is then used to generate better estimates of the image plane phasorgrams and so on. An index of the goodness of these estimates is the squared difference between the estimated and measured amplitude of each pixel summed over all the phasorgrams. Dividing this number by the sum of the squared measured amplitudes of each pixel over all pixels over all phasorgrams (the defined phasorgrams' energy) gives a normalized index called the fractional error. Note that, the smaller the fractional error, the more the phasorgram estimates look like the measured phasorgrams and presumably the better the complex function estimate resembles the sought function.
FIG. 2 shows the way in which fractional error decreased as a function of iteration number. The curve is monotonic decreasing, which will be shown to be a necessary consequence of the algorithm used to process the data. During the 37 iterations required for this problem, the normalized mean squared error defined in Kim's paper, which is the same as our fractional error except for a constant factor, decreased by four orders of magnitude. FIG. 3 shows contour plots of the way in which the sought phase distribution developed for iterations 1, 10, 20, 30 and 37. The region of support (the area in the plane in which the wave amplitude is not close to zero) in the initial plane was quite small, making it a relatively easy problem to solve. Other details of this simulation are that the field of pictures and kinoform lenses was 64 by 64 pixels for a total of 4096 pixels. The kinoform lenses had phase elements measuring 2 by 1 pixels. Each of these 2 pixel units multiplied the initial plane wave by either 1.0 or −1.0. They were distributed to abutting areas of the kinoform lenses on a coin-toss basis.
A second demonstration solved a more difficult problem in which, once again, the measured unperturbed image was constant. However, the unknown phase distribution was totally random from 0 to 2π on a pixel-by-pixel basis in a field measuring 64 by 64 pixels. The region of support in both conjugate planes was the entire plane. The sequentially placed perturbing devices in the initial plane were 15 holed stops. The holes in each of the 15 stops measured 5 pixels by 5 pixels. They were distributed in each stop to abutting areas on a coin toss basis. FIG. 4 shows a typical holed stop.
This particular demonstration took 237 iterations to reach a fractional error of less than 1 in 104, providing an essentially perfect phase retrieval. The random nature of the phase in this problem makes it difficult to appreciate the progress of the algorithm. However, the phase distribution estimated after the first and the 225th iteration are shown contoured in FIG. 5. The phase distribution on the 225th iteration is essentially the distribution sought. Note the detail which has been reconstructed. FIG. 6 shows the corresponding curve of fractional error as a function of iterations. It is perhaps worthwhile to compare this curve with that of FIG. 2.
A third demonstration showed the results of using holed stops in an initial plane containing 8 pixel by 8 pixel repeat units in a field of 64 pixels by 64 pixels. The amplitudes and phases of each of the 64 pixels in a repeat unit were chosen at random, the amplitudes between 0 and 2 and the phases between 0 and 2π. The holed stops were designed according to the same random scheme outlined above with the holes being 5 pixels by 5 pixels. The amplitude distribution of this demonstration is shown as FIG. 7 and the new algorithms convergence to a fractional error of less than 1 part in 108 is shown in FIG. 8. This run was obviously planned with crystallography in mind and the result once again was essentially perfect. Nine phasorgrams were used.
Thus, with the new information generated in the observation plane as very distinct pictures or phasorgrams for each known perturbation device placed in the initial plane, it appears that we can solve the phase retrieval problem uniquely. This remark is qualified because although we can and will show that the error must decrease with each iteration or at worst remain the same, no proofs have been devised about uniqueness or stagnation despite the fact that they have not been a problem in dozens of trial simulations. Each of these simulations has been solved using more or fewer phasorgrams with there being no surprises in the reconstructed complex wavefunctions. Stagnation (the error remaining the same from iteration to iteration) has not occurred as long as the number of phasorgrams was sufficient. However, we have yet to devise a rule for deciding the minimum number of phasorgrams required for a given problem. Fienup and Wackerman [4] dealt with the problem of stagnation in many of the variant algorithms of the GS method. It is noted that these algorithms deal with only two pictures.
FIG. 9 is a flow chart of the new algorithm. Its process can be followed by beginning with the N phasorgrams, which are amplitude distributions of complex wavefunctions. One seeks the unique phase distribution which is valid for each of them. To begin, assume that there is no reason to estimate one possible phase distribution over another. Set the phase distribution for all the phasorgrams to zero. Therefore, the first estimate is that they are all real functions. Now Fourier transform each of them and retain the N complex transform functions. If the perturbing device associated with a particular phasorgram was a stop, then all the pixels on each transform that were stopped have their phasors set to zero amplitude. If the perturbing devices were optical lenses or kinoform lenses, then remove the phase effect of the perturber. That is, if the perturber advanced a pixel phasor by 1 rad, then retard that pixel phasor by 1 rad. Carry out the same removal procedure for all the pixels in all the transform functions. Now, on a per pixel basis, add together vectorially the phasors for the same pixel from each of the N transform complex functions (initial wave-front estimates) and average them. Phasors whose amplitude is zero because they were blocked are not considered in the average. Thus, if there were 10 transforms and for a particular pixel, in three of the transforms, that pixel was blocked, then the sum of the remaining seven phasors is divided by seven. This rule does not apply if the perturbers are lenses or kinoforms. There, zero-amplitude phasors are counted as values in the averaging. The resulting single averaged transform is the new estimate of the initial planewave front. Now apply to this wave estimate the effect of each different perturber in turn. Then inverse Fourier transform each of the filtered initial wave estimates and retain them. Each of these new complex functions is an uncorrected estimate of the phasorgram of its respective filter or perturber. Correct this estimate by setting the amplitude of each pixel phasor in each of the new phasorgram estimates to the measured amplitude value while retaining the phase of the estimate. Now the next iteration cycle begins with these new phasorgram estimates.
Note what has been done here. There were N phasorgrams and complete knowledge of the N perturbers which created them. No information was in hand about perhaps the limited support of the initial wavefunction or about whether it was a complex function or not and so on as was the case in the paper by Kim. Instead, this algorithm relied on the principle of error energy reduction exclusively. What is error energy and how is it used here? Its definition and the way that it is used become clear as the processing algorithm is studied. The algorithm begins by estimating the complex function in the plane of observation for each of the N phasorgrams. The phasor for each pixel in all N phasorgrams will have its amplitude set to the measured amplitude and for convenience the phase of each phasor in set to zero. Each of the N phasorgrams is Fourier transformed and the effects of their corresponding perturbers are removed. This yields N different estimates of the complex initial wave. These N estimates must now be combined to yield a single initial wave estimate according to some principle: the error reduction principle. Each initial wave pixel estimate has N or fewer phasor estimates and each of these will be changed to the same phasor estimate for that pixel. That change will require the vector addition to each of these phasor estimates of a different vector to change it to the single final phasor estimate for that pixel. We seek a method of selecting the final phasor estimate in such a way that the sum of the square of the norm of each of the distinct vectors, which are added to each of the N or fewer phasor estimates to yield the final single phasor estimate for the pixel, is a minimum. The sum of this number taken over all the pixels in each of the N initial wave estimates will be called the correction energy. One wants the correction energy to be as small as possible. Accepting this requirement for the moment, the question becomes: how is this minimum correction energy achieved?
Consider a single pixel with its say L phasor estimates. Let E be the total correction energy contribution of this pixel. Then one has the following equation:
                              E          =                                                    ∑                                  i                  =                                      1                    ⁢                                                                                  ⁢                    to                    ⁢                                                                                  ⁢                    L                                                              ⁢                                                (                                                            U                      i                                        -                                          U                      0                                                        )                                2                                      +                                          (                                                      V                    i                                    -                                      V                    0                                                  )                            2                                      ,                            (        2        )            where (Ui, Vi) is the ith phasor estimate's real and imaginary coordinates and (Uo, Vo) is the final phasor estimate's real and imaginary coordinates. Expanding yields
                    E        =                                            ∑                              i                =                                  1                  ⁢                                                                          ⁢                  to                  ⁢                                                                          ⁢                  L                                                      ⁢                          U              i              2                                -                      2            ⁢                          U              i                        ⁢                          U              0                                +                      U            0            2                    +                      V            i            2                    -                      2            ⁢                          V              i                        ⁢                          V              0                                +                                    V              0              2                        .                                              (        3        )            U and V are independent variables; so solving for each separately yields
                                          ⅆ            E                                ⅆ                          U              0                                      =                                                            ∑                                  i                  =                                      1                    ⁢                                                                                  ⁢                    to                    ⁢                                                                                  ⁢                    L                                                              ⁢                                                -                  2                                ⁢                                  U                  i                                                      +                          2              ⁢                              U                0                                              =          0                                    (        4        )            or
                                                        1              L                        ⁢                                          ∑                                  i                  =                                      1                    ⁢                                                                                  ⁢                    to                    ⁢                                                                                  ⁢                    L                                                              ⁢                              U                i                                              =                      U            0                          ;                            (        5        )            similarly
                                          1            L                    ⁢                                    ∑                              i                =                                  1                  ⁢                                                                          ⁢                  to                  ⁢                                                                          ⁢                  L                                                      ⁢                          V              i                                      =                              V            0                    .                                    (        6        )            That is, averaging the phasor estimates provides the final phasor estimate that was sought. This procedure only guarantees an extreme but the physical situation is such that there can be no maximum.
Now, in turn each of the perturbers are applied to this estimate and inverse Fourier transformed to yield N new estimates of the N phasorgrams in the observation plane. The perturbers (either optical lenses or kinoform lenses or holed stops) have been chosen to maintain the energy (norm squared) of both the initial uncorrected waveform estimates and the correction functions that were added to them. The linearity of the Fourier transform ensures that the new estimates of the complex phasorgrams will be composed of two parts; the previous iteration estimate plus the transform of the respective correction functions. Furthermore, Parseval's theorem insures that the energy of each of these two parts of a phasorgram estimate will have the same energy as it did in the initial plane. Simply put, Parseval's theorem states that the energies (the integrated squared norm of a complex function) of a function in both of the two conjugate Fourier planes will be the same. One now chooses to make the correction to each pixel phasor in the observation plane which is the smallest possible. Referring to FIG. 10, the Argand diagram shows that by retaining the phase of the new estimate and replacing the phasor amplitude with the known amplitude the new error vector CB, in the observation plane, on a pixel-by-pixel basis will have the same or less energy than the correction energy vector AC from the initial plane. The total error energy over all phasorgrams as the process proceeds through the Fourier transforms and the removal of the perturber effects to the N new estimates of the initial wave front is the upper limit of the total correction energy. That is because the new estimate of the initial wave is the same as the previous estimate of the initial wave plus the transform of the error function that was added in the observation plane. Parseval's theorem ensures that that error function energy is the same in both conjugate planes. Thus, at this point, the algorithm could return to the previous initial wave estimate, causing the correction function energy to equal the error energy and the algorithmic process would stagnate. Neither the correction energy nor the error energy would change. However, by averaging the phasor estimates, one is guaranteed to have the lowest total correction energy possible. In this way, the algorithm proceeds from one conjugate plane to the other reducing the correction energy in the initial plane and then reducing the error energy in the observation plane. The goal of course is to find a phase distribution which makes the error energy or the correction energy zero or as close to zero as possible.
A slightly different situation occurs when there are measurements of the amplitude distribution not only in the observation plane (the phasorgrams) but also in the initial plane. The phasor estimates must be averaged as in the above algorithm and then, retaining the phasor angle, the amplitude of the averaged phasor is adjusted to match that measured. Thus, if the measurement of the amplitude of the wave in the initial plane is in hand, then with this adjustment the algorithm will be much more efficient.
A new method of phase retrieval has been documented. It rests on the diffraction equation of Raleigh and Sommerfeld and also of Kirchhoff, Fresnel and Huygens as simplified by the paraxial approximations and the Fresnel diffraction requirements. This equation is given as equation (1). Simplified, the problem essentially became the following: given the amplitude of a complex two-dimensional function in either or both of its conjugate Fourier domains, is it possible to discover the phase function uniquely and reliably? The answer was no, based on those measurements alone. It was shown, however, that, with experimentation, additional measurements could be made to provide the answer to the phase retrieval problem. For example, in an ordinary camera, the light wave in the back focal plane of the lens shares the diffraction equation (1) relationship with the imaging plane. Both planes are available and by using optical lenses or diverse stops or kinoform lenses in the back focal plane, new pictures (phasorgrams) may be made in the imaging plane. An indefinite number of these phasorgrams plus precise knowledge of the initial plane perturbers (lenses, etc.) that produced them is enough additional information to ensure phase retrieval. Similarly, diffractograms taken from a crystal specimen in which diverse stops were laid against the side of a radiating crystal could be solved.
These data powered a new iterative algorithm to the solution of the phase retrieval problem. The algorithm was elucidated and it was found that its defined error energy must decrease with each iteration or at worst remain the same. It was demonstrated that, when the error was zero or very small, the answer was in hand. However, it still remains to be proved that this necessarily follows. Perturber designs remain to be opitimized and the trade-off between the designs and the number of phasorgrams required to solve the phase retrieval problem needs to be studied. Simulations have been carried out but it still remains actually to build a device that would employ these principles.